## On the solution set of evolution inclusions driven by time dependent subdifferentials.(English)Zbl 0810.34059

The author considers the following evolution inclusion of subdifferential type: $-x'(t)\in \partial\varphi(t,x(t))+ F(t,x(t))$ almost everywhere in $$[0,b]$$ with the initial condition $$x(0)= x_ 0$$. The function $$x\to \varphi(t,x)$$ is convex, $$\partial\varphi(t,x)$$ is the subdifferential of $$\varphi(t,\cdot)$$ at $$x$$ and $$(t,x)\to F(t,x)$$ is the set-valued perturbation of the problem. Two existence results are presented: one for the case when the multifunction $$F$$ is nonconvex valued and the other for $$F$$ convex valued. Then, he compares the solution sets of those two problems, establishing a relaxation result. An example of a nonlinear parabolic optimal control problem is presented.

### MSC:

 34G20 Nonlinear differential equations in abstract spaces 49J52 Nonsmooth analysis 49K27 Optimality conditions for problems in abstract spaces 34A60 Ordinary differential inclusions 47N20 Applications of operator theory to differential and integral equations